I'm buying some stuff from elexp.com and I thought I'd pick up some heat shrink tube. I can choose from sizes 1/16", 1/8", 3/16", 1/4", 3/8" and 1/2". I want to buy 2 or 3 different sizes - which ones should I get (that I'll use the most often)?

Yes, do keep in mind that it has about the same shrink amount regardless of size... so it therefore shrinks MORE on smaller sizes relative to wire size. What I'm saying is that you likely want a larger variation of larger sizes and smaller variation of smaller sizes... just more of it.

Make it as small as you can to fit over the wire. The larger stuff is used when combining bunches of wires to make a multicored connection.There is a limit on how much it shrinks so don't get it too big.

There really isn't a lot of use for heat-shrink tubing with Arduino projects.

I suppose that depends on the project. Or, maybe, how fond you are of excessive fiddling.

That's both ends of a DS18B20, on CAT5. I just like to have things nice and tidy. Also, I'll use it to color-code wires, as in this case, red = +V and black = GND. Also, all my wired-up sensors have a self-stick label on them, with the sensor address. Those aren't sticking quite so well, so I'll probably put clear heat-shrink over them.

... it is poor civic hygiene to install technologies that could somedayfacilitate a police state. -- Bruce Schneier

a 22g wire with (typical) insulation is just barely less than 1/16 inch, which makes 1/16 inch heat shrink a bit too small and 1/8 a bit too large. 3/32 would be better. Now, if you want to cover splices or solder connections to 22g wire, then 1/8 is probably the right size...